Notes, summaries, assignments, exams, and problems for Mathematics

Quick Graph Identification

• Holes: Represented by open circles.
• Vertical Asymptotes: Represented by dashed vertical lines.
• Horizontal or Slant Asymptotes: Represented by dashed lines.
• Note: The graph approaches but never touches the asymptotes.

Understanding Domain

• Exclude values that make the denominator equal to zero.
• Even if a factor cancels, the value is still excluded from the domain.

Example: (x + 3) / (x(x + 3))
Domain: x ≠ -3, x ≠ 0

Final One-Pass Checklist

1. Factor and cancel.
2. Find holes.
3. Find vertical asymptotes.
4. Find x-intercepts.
5. Find the y-intercept.
6. Find horizontal or slant asymptotes.

Exponential and Logarithmic Functions

Exponential: f(x) = a · b^(x - h) + k (where b > 0, b ≠ 1)
Logarithmic: f(x) = a · log_b(x - h) + k
Note: Logarithms... Continue reading "Mastering Rational, Exponential, and Logarithmic Functions" »

Algebraic Operations and Equations

Completing the Square

To complete the square, take 1/2 of b, then square it. When a ≥ 1, remember to factor out the a first.

Exponents and Radicals

• Negative Exponents: Take the reciprocal of the base and make the exponent positive. In fractions, move the negative exponent to the bottom and make it positive.
• Radicals: Always check your answers.
• Dividing Radicals: Divide the numbers inside by the other inside, and divide the outside numbers by the other outside numbers.
• Radical Exponents: Only square or cube the number/variable under the radical unless there are parentheses.
• Conversion: To convert a radical into an exponent:
• Rationalizing: To get a radical off the bottom, multiply by the radical over itself (so it

Executive Summary of Regression Models

• Simple Linear Regression: On average, for every 1-unit increase in [X], the expected [Y] changes by β1 units (95% CI: …).
• Multiplicative Model: On average, a 1-unit increase in [X] multiplies the median [Y] by exp(β1), resulting in a 100·(exp(β1)–1)% change (95% CI: …).
• Power Law/Elasticity: A 1% increase in [X] is associated with a β1% change in [Y] (95% CI: …).
• Categorical Variable: Students in Group A scored on average β1 units higher or lower than those in Group B (95% CI: …).
• Categorical Variable (3-Group): After adjusting for [X], students taught with Method 2 scored on average β1 units higher than those with Method 1; Method 3 scored β3 units lower.
• Interaction: For Group A, a 1-unit

Key Concepts in Engineering Economics

Engineering Economics is the science dealing with quantitative analysis techniques for selecting the most preferable alternative from several technically viable options.

Fundamental Principles

Four fundamental principles must be applied in all engineering economic decisions:

• The time value of money
• Differential (or incremental) cost and revenue
• Marginal cost and revenue
• The trade-off between risk and reward

Core Terminology Explained

Ethics

A set of principles that guides a decision-maker in distinguishing between right and wrong.

Market Interest Rate

The interest rate quoted by financial institutions, which refers to the cost of money for borrowers or the earnings from money for lenders.

Interest Rate

The cost, or price,

Creating a comprehensive cheat sheet for statistical analysis in a Bachelor of Technology (BTech) program involves covering a wide range of topics and formulas. Below is a condensed version of this essential reference:

Descriptive Statistics

Measures of Central Tendency

• Mean: \(\bar{x} = \frac{\sum{x}}{n}\)
• Median: Depends on data arrangement.
• Mode: Most frequent value.

Measures of Dispersion

• Variance: \(s^2 = \frac{\sum(x - \bar{x})^2}{n-1}\)
• Standard Deviation: \(s = \sqrt{s^2}\)
• Range: \(Range = Max(x) - Min(x)\)
• Interquartile Range (IQR): \(Q3 - Q1\)

Measures of Shape

• Skewness
• Kurtosis

Probability

Basic Probability

• \(P(A)\) = \(\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\)
• \(P(A \cup B)\) = \(P(A) + P(B) - P(A \cap B)\)

Conditional

Managerial Decision-Making and Business Analytics

Types of Managerial Decisions

To effectively plan, coordinate, and lead, managers make several types of decisions:

• Strategic Decisions: Address high-level issues and the overall direction of the organization. They define future goals and are long-term and complex.
• Tactical Decisions: Focus on how to achieve the goals and objectives set by the strategy. These are typically made by mid-level management for the medium term.
• Operational Decisions: Pertain to day-to-day operations. They are made by operations managers and are often simple and routine.

The Decision-Making Process (DMP)

A structured approach to decision-making involves several key steps:

1. Identify and define the problem.
2. Determine the criteria

Understanding Bookkeeping

Bookkeeping is the systematic recording of financial transactions of a business in books of accounts on a day-to-day basis.

Objectives of Bookkeeping

• Systematic Record: To keep a complete and permanent record of all business transactions.
• Ascertain Profit or Loss: Helps in finding profit or loss at the end of the accounting period.
• Ascertain Financial Position: Helps in knowing assets and liabilities of the business.
• Legal Evidence: Acts as proof in legal matters.

Advantages of Bookkeeping

• All transactions are properly recorded.
• Management can take better decisions.
• Provides information about profit, loss, assets, and liabilities.
• Makes auditing easier.
• Helps compare past and present performance.

Accounting Fundamentals

Accounting... Continue reading "Bookkeeping and Accounting Fundamentals Explained" »

Handshaking Lemma

In any undirected graph, the sum of the degrees of all vertices is twice the number of edges.

Formula: Σdeg(v) = 2|E|

Euler's Formula for Planar Graphs

For planar graphs, the relationship between vertices (V), edges (E), and regions (R) is defined as:

V - E + R = 2

Sum of Degrees and Odd Vertices

The sum of the degrees of all vertices in any graph is always even because each edge contributes 2 to the total sum. Furthermore, the number of vertices with an odd degree must always be even.

Graphs with No Odd Degree Vertices

If all vertices in a graph have an even degree, the graph is Eulerian, meaning it contains an Eulerian circuit.

Complete Graphs

A complete graph with n vertices, denoted as K_n, has an edge between every pair of distinct... Continue reading "Essential Graph Theory Formulas and Concepts" »

Understanding Quartiles

Quartiles divide a set of ordered data into four groups with equal numbers of values. The three dividing points are Q1, the median (Q2), and Q3.

• Interquartile Range (IQR): Defined as Q3 – Q1, this represents the range of the middle half of the data. It provides a measure of spread by showing how closely the data are clustered around the median.
• Semi-interquartile Range: One half of the interquartile range.

Quartile Formulas

• Q2 (Median): (n+1) / 2
• Q1: (n+1) / 4
• Q3: 3(n+1) / 4

Outlier Formula

To identify outliers, use the following boundaries:

• Left Boundary: Q1 - 1.5(IQR)
• Right Boundary: Q3 + 1.5(IQR)

Deviation and Standard Deviation

Deviation tells you how far a single data value is from the mean (the difference between a value... Continue reading "Understanding Quartiles, Standard Deviation, and Percentiles" »

1. Type I and Type II Errors

Type I Error (α): Occurs when a researcher rejects a true null hypothesis (a "false positive").

Type II Error (β): Occurs when a researcher fails to reject a false null hypothesis (a "false negative").

The goal of statistical testing is to minimize both errors simultaneously.

2. Parametric vs. Non-Parametric Statistics

Parametric Tests: These assume data is normally distributed and use interval/ratio scales (e.g., t-test, ANOVA).

Non-Parametric Tests: These are "distribution-free" tests used for nominal/ordinal data or small samples (e.g., Chi-square, Mann-Whitney U).

Parametric tests are generally more powerful if their assumptions are met.

3. Null Hypothesis (H₀) vs. Alternative Hypothesis (H₁)

Null Hypothesis (H₀)